Let's do math again.
Your transformer is a 24V transformer, right? At 24V and 2 ohms resistance, you'll push 12A through that wire. P=VI=I^2*R; that's 288W(!), which is more (by a long shot) than your average soldering iron uses, AND 12 times the rated current of that transformer. In fact, I daresay it's a good thing that your fuse went, or you'd have had something really, REALLY bad happen (fire? Melted wire? Who can say what the weakest link would have been?).
Even if you put the rectifier in, you're only killing half the power, and very, very few rectifiers will work at 12A peak current. You'd need heatsinking at the very least.
So what DO you need to do? My 10W wire cut nicely- let's do some math based on 50% higher. P = I^2*R = V^2/R so for P = 15, I ~= 2.75A and V ~= 5.5V. I might consider using (as Theo suggested before
) an ATX power supply from a PC. One or another of the rails will need a load to pull the supply into regulation- it might be the 5V rail, in which case your wire provides the load quite nicely. If it's another one that's needed, well, Ax-man has a few high power resistors available, and I'm willing to be that if it's not the 5V supply that needs the load, it'd be the 3.3V supply.
A 6V supply might work okay- perhaps a battery charger for 6V hobby battery packs? Or 7.2V packs? That might work nicely- "quick chargers" for RC planes tend to charge at fairly high amperages, with little or no check on their current. In this case, you'd want to buy the cheapest one possible, as higher quality units would attempt to limit their charge current to preserve the life of the battery.
Here's a thought as to why the dimmer switch isn't a great solution- you want to adjust this voltage linearly, but that dimmer switch is set up to dim a logarithmic light source. Usually it's a cheat- they do two linear regions which nicely approximate a logarithmic scale. The reason for that is that light, like sound, is perceived to increase in a linear fashion when its intensity increases logarithmically. For this, though, you want a linear increase- the logarithmic increase would give you little to no increase in temperature for a while, then the temperature would suddenly skyrocket.